Problem 54 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int(1.3)^{x} d x$$

Step-by-Step Solution

Answer

The most general antiderivative is \(\frac{(1.3)^x}{\ln(1.3)} + C\).

Step 1: Identify the Problem

We are tasked with finding the most general antiderivative of the function \((1.3)^x\). This involves integrating \((1.3)^x\) with respect to \(x\).

Step 2: Substitution Consideration

Notice that the function \((1.3)^x\) can be written using the base change formula as \(e^{x \ln(1.3)}\). This suggests using a substitution to simplify our integration.

Step 3: Substitute Function Expression

Let \(u = x\ln(1.3)\). Then, \(du = \ln(1.3) \, dx\), which implies \(dx = \frac{1}{\ln(1.3)} \, du\). Substituting these into the integral gives \(\int e^u \, \frac{1}{\ln(1.3)} \, du\).

Step 4: Integrate Using Substitution

The integral \(\int e^u \, du\) is easily evaluated to be \(e^u + C\). Therefore, the integral becomes \(\frac{1}{\ln(1.3)} e^u + C\), where \(C\) is the constant of integration.

Step 5: Re-substitute to Original Variables

Substitute back \(u = x\ln(1.3)\) into the expression: \(\frac{1}{\ln(1.3)} e^{x\ln(1.3)} + C \). Since \(e^{x\ln(1.3)} = (1.3)^x\), the result becomes \(\frac{(1.3)^x}{\ln(1.3)} + C\).

Step 6: Verification by Differentiation

Differentiate \(F(x) = \frac{(1.3)^x}{\ln(1.3)} + C\) with respect to \(x\) to check the solution. By using the chain rule, the derivative is \((1.3)^x\), confirming that our antiderivative is correct.

Key Concepts

Antiderivative

The concept of an antiderivative, also known as an indefinite integral, is foundational in calculus. It involves finding a function whose derivative matches the original function we started with. In our exercise, we aim to find the antiderivative of the function \((1.3)^x\). This means identifying a function \(F(x)\) such that \(F'(x) = (1.3)^x\). Antiderivatives encompass a broad range of possibilities. Generally, any function \(F(x) + C\) is a valid antiderivative where \(C\) is a constant. This is due to the fact that the derivative of a constant is zero, hence it doesn't affect the original function after differentiation. In the exercise, we found the antiderivative \(\frac{(1.3)^x}{\ln(1.3)} + C\) as our most general solution.

Substitution Method

The substitution method is a powerful tool for simplifying complex integrals. It involves transforming a function into an easier form by introducing a new variable \(u\). For our integral of \((1.3)^x\), the expression is rewritten using exponential terms, allowing easier manipulation.**Steps for the substitution method:**

• Choose a substitution: Set \(u = x \ln(1.3)\).
• Express \(dx\) in terms of \(du\): Calculate \(du = \ln(1.3) \, dx\) leading to \(dx = \frac{1}{\ln(1.3)} \, du\).
• Substitute all occurrences in the integral: The original integral \(\int (1.3)^x \, dx\) becomes \(\int e^u \, \frac{1}{\ln(1.3)} \, du\).
• Solve the simpler integral: Now, you can integrate \(e^u\) easily!

This substitution simplified the process and allowed us to integrate and then change back to the original variable \(x\).

Integration by Parts

Although not directly used in our specific problem, integration by parts is another vital technique in calculus. It is especially useful for integrating products of functions. This technique is grounded in the product rule for differentiation and follows the formula:\[ \int u \, dv = uv - \int v \, du \]**When to use integration by parts:**

• When facing an integral of a product of functions.
• When substitution does not simplify the integral substantially.
• Common in functions such as polynomials multiplied by exponential or trigonometric functions.

Understanding when to employ integration by parts can be immensely beneficial for tackling integrals that are not easily simplified by substitution.

Differentiation Verification

After finding an antiderivative or indefinite integral, it's critical to verify the solution. Differentiation verification involves taking the derivative of your antiderivative result to ensure it matches the original function from the integral.In the given problem, the derived antiderivative is \(F(x) = \frac{(1.3)^x}{\ln(1.3)} + C\). To verify:

• Differentiate \(F(x)\) with respect to \(x\): Apply the differentiation rules to get \(\frac{d}{dx} \left( \frac{(1.3)^x}{\ln(1.3)} \right)\).
• Use the chain rule: The derivative of \((1.3)^x\) is \((1.3)^x \ln(1.3)\). So, after simplifying, you should return to the original function \((1.3)^x\).
• Confirmation: If the differentiated result matches the original integrand, then your antiderivative is correct!

Differentiation verification serves as a robust checkpoint to ensure integration accuracy, confirming that no errors have been made during the process.